In the above analysis, we have ignored the GW emission, which would not affect the evolution substantially [1]}. The critical angular momentum \(a_{*C}^{nlm}\)  could be estimated with Eq. (REF ). The critical BH mass \(M_C^{nlm}\)  at which the quasi-normal phase starts could also be estimated by setting \(a_{*C}^{nlm} = 1\)  in Eq. (REF ). The accelerating and decelerating phases are separated by the time with \(\dot{a}_*=0\) . At this time, the \(\dot{J}\)  is still positive. A detailed analysis of these two quantities is given in App. .
